Let's consider that $$4x^2+9y^2+16z^2+12xy+16xz+24yz+2x+4y+6z+1=0$$. Classify the quadric.
Development:
The matrix associated with the equation of the quadric is: $$$\overline{A} = \begin{bmatrix} 4 & 6 & 8 & 1 \\ 6 & 9 & 12 & 2 \\ 8 & 12 & 16 & 3 \\ 1 & 2 & 3 & 1 \end{bmatrix}$$$ We are going to calculate, now, its euclidean invariants. $$$det(x \cdot I-\overline{A})=x^4-30x^3+15x^2+6x$$$ $$$det(x \cdot I - A)=x^3-29x^2$$$ Therefore, we have : $$$\left \{ \begin{array}{l} D_4=0 \\ D_3=-6 \\ D_2=15 \\ D_1=30\end{array} \right.$$$ $$$\left\{ \begin{array}{l} d_3=0 \\ d_2=0 \\ d_1=29 \end{array} \right.$$$
The index of the quadric is $$0$$ due to the fact that the condition $$d_1\cdot d_3 < 0$$ and $$d_2 < 0$$ is not satisfied.
Solution:
As $$D_4=0, d_3=0, d_2=0, D_3=-6$$, for the classification algorithm we can see that it is a parabolic cylinder.